Complex network analyses of functional connectivity have consistently revealed non-random (modular, small-world, scale-free-like) behavior of hard-thresholded networks constructed from the right-tail of the similarity histogram. In the present study we determined network properties resulting from edges constrained to specific ranges across the full correlation histogram, in particular the left (negative-most) tail, and their dependence on the confound signal removal strategy employed. In the absence of global signal correction, left-tail networks comprised predominantly long range connections associated with weak correlations and were characterized by substantially reduced modularity and clustering, negative assortativity and I? < 1 Deconvolution of specific confound signals (white matter, CSF and motion) resulted in the most robust within-subject reproducibility of global network parameters (ICCs~0.5). Global signal removal altered the network topology in the left tail, with the clustering coefficient and assortativity converging to zero. Networks constructed from the absolute value of the correlation coefficient were thus compromised following global signal removal since the different right-tail and left-tail topologies were mixed. These findings informed the construction of soft-thresholded networks, replacing the hard thresholding or binarization operation with a continuous mapping of all correlation values to edge weights, suppressing rather than removing weaker connections and avoiding issues related to network fragmentation. A power law adjacency function with I? = 12 yielded modular networks whose parameters agreed well with corresponding hard-thresholded values, that were reproducible in repeated sessions across many months and evidenced small-world-like and scale-free-like properties.